Finite Group Databasebeta

← #13331788

Subgroups — #13331788

Order 1792.

812 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3556C211yesmaximal, characteristic
2#11740819C212yesmaximal
3#11740784C211yesmaximal, characteristic
4#11729692C214yesmaximal, complemented
5#11729691C214yesmaximal, complemented
6#11729588C211yesmaximal, characteristic
7#11729580C212yesmaximal
8C7×#564—22no—
9C7×#557—22no—
10C7×#557—22no—
11C7×#557C2214yes—
12C7×#541—24no—
13C7×#541—22no—
14C7×#533C2212yes—
15C7×#533C2211yescharacteristic
16#90161C2211yescharacteristic
17#89866—22no—
18#89862—22no—
19#89851C2212yes—
20#89850C2212yes—
21#89836—22no—
22#89824—24no—
23#89803C2214yes—
24#89792C2214yes—
25#89770—22no—
26#89696C2212yes—
27C2×C4×C7⋊C8C2211yescharacteristic
28#89342C2214yes—
29Q8⋊(C7⋊C8)C2214yes—
30Q8⋊(C7⋊C8)C2214yes—
31#18703—71nomaximal, Sylow
32C7×C4⋊Q8—22no—
33C7×C4⋊Q8—24no—
34C7×C4⋊Q8D412yes—
35C7×C4⋊Q8D412yes—
36C7×C4⋊Q8C2312yes—
37C7×#126—28no—
38C7×#125—22no—
39C7×#125—28no—
40C7×#125—28no—
41C7×#125—24no—
42C7×#125—28no—
43C7×#125—22no—
44C7×#125D412yes—
45C7×C22⋊Q8—22no—
46C7×C22⋊Q8—22no—
47C7×C22⋊Q8D414yes—
48C4×C7×Q8—24no—
49C4×C7×Q8—24no—
50C4×C7×Q8—24no—
51C4×C7×Q8—22no—
52C4×C7×Q8—22no—
53C4×C7×Q8C2314yes—
54C7×#117—22no—
55C7×#117D412yes—
56C2×C42×C7C2311yescharacteristic
57C2×#1144—24no—
58#6489—21no—
59#6487—22no—
60C2×#1136—22no—
61C22×C7⋊C8—21no—
62#6433—21no—
63#6432C2312yes—
64C4×Dic14—22no—
65#6409—24no—
66#6384—24no—
67#6382—24no—
68#6382—24no—
69#6381—24no—
70C4⋊(C7⋊C8)—22no—
71C4⋊(C7⋊C8)C2314yes—
72C4×C7⋊C8—21no—
73C4×C7⋊C8C2311yescharacteristic
74C4×C7⋊C8C2311yescharacteristic
75#3556D711yescharacteristic
76#3255—72no—
77#3160—71no—
78#1653—74no—
79#1648—74no—
80#1561—71no—
81#1493—72no—
82C2×C7×Q8—22no—
83C2×C7×Q8—22no—
84C2×C7×Q8C2×D414yes—
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7C2×D412yes—
87C2×Dic14—22no—
88C7×C4⋊C4—28no—
89C7×C4⋊C4—28no—
90C7×C4⋊C4—28no—
91C7×C4⋊C4—22no—
92C7×C4⋊C4—22no—
93C7×C4⋊C4—22no—
94C7×C4⋊C4—24no—
95C7×C4⋊C4—24no—
96C7×C4⋊C4—22no—
97C7×C4⋊C4—24no—
98C7×C4⋊C4C2×D414yes—
99C7×C4⋊C4C2×D414yes—
100C7×C22⋊C4—24no—
101C7×C22⋊C4—22no—
102C42×C7—28no—
103C42×C7—22no—
104C42×C7—24no—
105C42×C7C2×D412yes—
106C42×C7C2×D411yescharacteristic
107C42×C7C2411yescharacteristic
108C22⋊Dic7—41no—
109#1144—84no—
110C4⋊Dic7—22no—
111C4⋊Dic7—22no—
112Dic7⋊C4—42no—
113C4×Dic7—41no—
114#1136—82no—
115C2×C7⋊C8—24no—
116C2×C7⋊C8—21no—
117C2×C7⋊C8—21no—
118C2×C7⋊C8—24no—
119#564—22no—
120#557—22no—
121#557—22no—
122#557C2×D714yes—
123#541—24no—
124#541—22no—
125#533—71no—
126#533C2×D712yes—
127#533C2×D711yescharacteristic
128#500—72no—
129#499—142no—
130#498—72no—
131#475—74no—
132#475—142no—
133#470—144no—
134#462—74no—
135#462—142no—
136#427—142no—
137#423—72no—
138C2×C4×C8—71no—
139#332—74no—
140Q8⋊C8—74no—
141Q8⋊C8—74no—
142C23×C7#12011yescharacteristic
143C7×Q8—24no—
144C7×Q8—24no—
145C7×Q8—42no—
146C7×Q8—42no—
147C7×Q8—44no—
148C2×C4×C7—22no—
149C2×C4×C7—22no—
150C2×C4×C7—22no—
151C2×C4×C7—24no—
152C2×C4×C7—24no—
153C2×C4×C7—24no—
154C2×C4×C7#12012yes—
155C2×C4×C7#12011yescharacteristic
156C2×C4×C7C22×D411yescharacteristic
157C2×C4×C7C22×D412yes—
158C2×Dic7—42no—
159Dic14—42no—
160Dic14—42no—
161C7⋊C8—22no—
162C7⋊C8—22no—
163C7⋊C8—44no—
164C2×Q16—144no—
165C2×M16—142no—
166C22×C8—141no—
167C4⋊Q8—22no—
168C4⋊Q8—24no—
169C4⋊Q8—72no—
170C4⋊Q8C22⋊D712yes—
171C4⋊Q8C22⋊D712yes—
172C4⋊Q8C22×D712yes—
173#126—28no—
174#125—22no—
175#125—28no—
176#125—28no—
177#125—24no—
178#125—28no—
179#125—22no—
180#125—141no—
181#125C22⋊D712yes—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—142no—
185C22⋊Q8C22⋊D714yes—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—142no—
192C4×Q8C22×D714yes—
193#117—22no—
194#117—141no—
195#117C22⋊D712yes—
196C2×C42C22×D711yescharacteristic
197#107—144no—
198#106—144no—
199#106—144no—
200C4⋊C8—74no—
201C4⋊C8—142no—
202#103—144no—
203#103—144no—
204C4×C8—71no—
205C4×C8—71no—
206C4×C8—141no—
207C22×C7—22no—
208C22×C7—21no—
209C22×C7C2×#12011yescharacteristic
210C4×C7—22no—
211C4×C7—21no—
212C4×C7—44no—
213C4×C7—44no—
214C4×C7—44no—
215C4×C7#45612yes—
216C4×C7#57712yes—
217C4×C7#57812yes—
218Dic7—82no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—142no—
222C2×Q8C2×C22⋊D714yes—
223C22×C4C2×C22⋊D712yes—
224C22×C4C2×C22⋊D711yescharacteristic
225Q16—564no—
226M16—562no—
227C2×C8—144no—
228C2×C8—141no—
229C2×C8—141no—
230C2×C8—144no—
231C4⋊C4—28no—
232C4⋊C4—28no—
233C4⋊C4—28no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—142no—
242C4⋊C4—142no—
243C4⋊C4—282no—
244C4⋊C4C2×C22⋊D714yes—
245C4⋊C4C2×C22⋊D714yes—
246C22⋊C4—24no—
247C22⋊C4—22no—
248C22⋊C4—281no—
249C42—28no—
250C42—22no—
251C42—24no—
252C42—281no—
253C42C2×C22⋊D711yescharacteristic
254C42C2×C22⋊D712yes—
255C42C23×D711yescharacteristic
256C2×C7—41no—
257C2×C7#301312yes—
258C2×C7C2×#45611yescharacteristic
259C23#651611yescharacteristic
260Q8—24no—
261Q8—24no—
262Q8—42no—
263Q8—42no—
264Q8—44no—
265Q8—282no—
266Q8—282no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—282no—
274C2×C4#651612yes—
275C2×C4#651611yescharacteristic
276C2×C4C22×C22⋊D711yescharacteristic
277C2×C4C22×C22⋊D712yes—
278C8—142no—
279C8—142no—
280C8—284no—
281C7#1870311yescharacteristic, complemented, Sylow
282C22—22no—
283C22—21no—
284C22C2×#651611yescharacteristic
285C4—22no—
286C4—21no—
287C4—44no—
288C4—44no—
289C4—44no—
290C4—562no—
291C4#9001812yes—
292C4#9052812yes—
293C4#9052912yes—
294C2—41no—
295C2#1174155912yes—
296C2C2×#9001811yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3556C2
socleC22×C7C2×#120
prime radicalC23×C7#120
hypercenter#3556D7
hypocenterC7#18703
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3556

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #11729692C2semidirect
5. #11729691C2semidirect
281. C7#18703semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C7×#35562
#117408192
#117407842
#117296922
#117296912
#117295882
#117295802
#187037